Estimator for an incidence rate While going through a statistics course for medicine students, I ran accross a problem related to incidence rates. The context of the problem is a chapter about the Poisson distribution. In the problem, 2300 smokers are followed over a 1 year span during which 24 of them develop lung cancer. They then want to compute the incidence rate of the process and proceed as follows:
$$\text{Incidence rate} = \frac{24}{2300-24/2}$$
At first, I didn't understand why they subtracted $24/2$, but I assumed it was some correction for the fact that since those 24 persons develop the cancer during the year, their time at risk is shorter than that of the ones not developing the disease. No further information was given in the textbook itself, at least not in the problem. A quick search confirmed that I'm thinking along the correct lines.
But I still don't understand the rationale for the formula. Can someone enlighten me? Also, if some references accessible to medical students could be given. I don't mind having more techical references as well.
 A: I propose modeling cancer occurrence as a Poisson process. Multiple events (appearance of tumors) are possible within the same individual over the time period of observation. If $\lambda$ is the rate of tumor appearance by year, the probability of 0 events is $e^{-\lambda}$, and the probability of 1 event or more is $p=1-e^{-\lambda}$.
You follow $n$ individuals during a year. The number of individuals with 1 event or more is $X \sim \mathrm{Bin}(n,p)$. The expected number is $E(X) = np = n(1-e^{-\lambda})$.
Now you observe $x$ events and want to estimate $\lambda$. First estimate $\hat p = {x\over n}$, then $\hat \lambda = - \log\left(1 - {x \over n}\right) \approx {x\over n} + {x^2 \over 2 n^2}$. By invariance of maximum-likelihood estimators, $\hat \lambda$ is the MLE of $\lambda$.
Your estimator is ${ x/n \over 1 - x/2n} \approx {x\over n} + {x^2 \over 2 n^2}$. The difference between the two estimators is about $x^3/6n^3$, which is very small if $x/n$ is small. I guess this provides some justification, even if some other modeling could possibly lead directly to your estimator.
A: Assuming diagnoses of cancer are uniformly spread across the year, the persons who are diagnosed are exposed to the risk of being diagnosed for (on average) half a year before that diagnosis.
Your link mentions the assumption of occurrence at the half-way point in the observation period but not where it comes from - which is just the assumption of uniformity. This assumption isn't always reasonable, and there are times when it can make a substantive difference. I'd recommend being aware of the assumption every time you use the formula, because you should consider its suitability and if it isn't suitable, whether it is likely to have a substantive impact on the estimate (in which case, a better assumption about the occurrence should be investigated)
